I'm referring to the Perl source make

Aaah ... however, I don't think the LINKTYPE=static argument does anything.

It's a long time since I've built perl statically, and I'm a bit hazy on the details. So I just tried a build of 5.26.0 configured with '-des -Uusedl -Dprefix=/home/me/perl526-static'.
However, that fails at the 'make' stage with:

libperl.a(regexec.o): In function `Perl__is_grapheme':
regexec.c:(.text+0x12b70): multiple definition of `Perl__is_grapheme'
lib/auto/re/re.a(re_exec.o):re_exec.c:(.text+0x1c020): first defined h
+ere
collect2: error: ld returned 1 exit status
makefile:369: recipe for target 'perl' failed
make: *** [perl] Error 1
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Have I missed something in my Configure args ? (Running 'make LINKTYPE=static' makes no difference.)
What is your 'perl -V' output ?
When I finish writing this, I'll try current blead - and send a post to p5p if it also fails to build statically.

Having had a read through the INSTALL file, it seems to me that what you've done should have worked ... but clearly hasn't wrt YAML::XS.
Do any of the other ext modules throw the same error when you try to load them ?
Try re-building YAML::XS from *fresh* source with:

perl Makefile.PL LINKTYPE=static
make
perl -Mblib -MYAML::XS -E 1
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Does that last command then run without error ?

UPDATE: Building perl statically from current git source (commit 97fcda7...) works fine for me, with all tests passing. For some reason 'make test' doesn't run the ext/YAML-LibYAML test suite
And I get the same results as rodd.
It makes no difference to the error whether YAML::XS is built with perl (in the ext directory) or built separately after the static perl has been built and installed.
I'm thinking that there's a bug in the way that YAML::XS is being dealt with. The INSTALL file does state this:

<quote>
If you unpack any additional extensions in the ext/ directory before running Configure, then Configure will offer to build those additional extensions as well. Most users probably shouldn't have to do this -- it is usually easier to build additional extensions later after perl has been installed. However, if you wish to have those additional extensions statically linked into the perl binary, then this offers a convenient way to do that in one step.
</quote>

UPDATE 2: There's a bit more to it than I expected. I was able to get a loadable YAML::XS installed as per follows:

cd YAML-LibYAML-0.65
perl Makefile.PL
make perl
make -f Makefile.aperl inst_perl MAP_TARGET=perl
make -f Makefile.aperl map_clean
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Cheers,
Rob